Rigid
Dualizing Complexes via Differential Graded Algebras
Amnon Yekutieli, BGU
Abstract: Rigid dualizing
complexes were introduced by Van den
Bergh in the context of
noncommutative algebraic geometry, where
they proved to be extremely
useful. The advantage of rigidity is
that it eliminates automorphisms,
thus making dualizing complexes
unique, and even functorial.
This talk is about rigid
dualizing complexes over *commutative*
K-algebras. If K is a field then
the "noncommutative" results
specialize to yield an enormous
amount of information on rigid
dualizing complexes and their
variance properties. Indeed, one
can recover most of the important
features of Grothendieck
duality (for affine schemes),
including explicit formulas, with
relatively little effort.
However we want to consider
commutative algebras over any
noetherian commutative base ring
K. It turns out that this causes
serious technical issues, due to
lack of flatness. Even defining
rigidity (i.e. writing Van den
Bergh's rigidity equation) is a
problem! Our solution was to use
differential graded algebras.
Thus, if A is a K-algebra which
is not flat, we replace A with a
quasi-isomorphic DG K-algebra
A^~ which has suitable flatness
properties, and use A^~ to
formulate the rigidity equation for
complexes of A-modules.
Actually, this method enables us
to work with relative rigid
complexes. Namely, given a
homomorphism A -> B between
K-algebras, we can consider rigid
complexes of B-modules relative
to A. (This is nontrivial even
when the base K is a field.) The
theory of rigid complexes we
developed is quite rich, and may be
of independent interest in ring
theory. In this talk I will
explain some of the features of
this theory.
When our base ring K is regular
(e.g. the ring of integers) we
obtain a comprehensive theory of
rigid dualizing complexes, once
again producing most of the
important features of Grothendieck
duality for affine K-schemes.
I'll discuss the main results, and
also some geometric consequences.
Full details can be found in the
preprint math.AG/0601654 at
http://arxiv.org.
This is joint work with James
Zhang (Univ. of Washington).